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Some background first.

I recently graduated (a couple of years ago) with a master's degree in applied mathematics. During graduate school I began working on a paper, which I continued to work on in my free time post-graduation. A complete working copy of the paper is done and I have posted it on the arXiv here. The work contained in the paper is completely original and solves an "open problem" (the problem is open but not anything necessarily revolutionary).

It was of my opinion that the paper contained publishable material. To verify, I emailed a professor at my alma mater with a copy of the current draft (current as of approximately six months ago). The professor did reply stating that the work was publishable and even suggested some Q1 and Q2 journals that might accept this type of work. While this was useful feedback, I received the reply in just a few days, so I doubt the professor in question had the ability to read my paper in depth.

The problem:

I have published a couple of papers before and thus have some experience in the world of academic publishing. That said, the scale and complexity of this paper is something I have never dealt with before so I am not comfortable with proceeding to publish it without help/guidance, i.e. on my own. In particular, I suspect I am going to have to divide the work into small portions and publish a few separate papers, but I don't know how best to do this and do not want to sink a lot of additional time into this without any direction. Also, the paper is very dense and I am concerned that its readability is not optimal. Given that I do not have a ton of experience and am essentially working in a vacuum, I also really desire to get feedback on the quality of my proofs, which I suspect are not as concise as they should be. My situation seems a little unusual and I suspect that the feedback I am looking for would typically be provided by an adviser in a Ph.D. program (not claiming the paper is worthy of a Ph.D.).

Given that I do not have an adviser that can provide detailed feedback, what should I do?

I have tried reaching out to researchers/experts with relevant backgrounds and offering authorship in exchange for the help I need. Since what I'm seeking entails a significant amount of work, this proposition seemed reasonable; however, my efforts have not lead to much fruit.

As professional mathematicians, how are these types of requests viewed and how might I go about reaching out for help? I do not have funding and so I considered the offering of authorship as a a reasonable incentive to get the help I need. Is this practice frowned upon and is there anything else I can do to increase my chances of getting a researchers attention?

This is a question migrated from SE academia, which I believe can be better addressed in this forum. I have modified the content to reflect my current circumstances. If this is considered an unacceptable practice I can remove this post.

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    $\begingroup$ To me (not knowing your field at all) your article on the arXiv strikes me more as a monograph (i.e. a book about a single topic) than an article. Perhaps that is a more appropriate form of publishing for this? $\endgroup$ Sep 14 at 15:52
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    $\begingroup$ I wonder if it might be appropriate to consider either the EMS Monograph series or the AMS Survey and Monograph series? $\endgroup$ Sep 14 at 15:59
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    $\begingroup$ Not knowing anything about this topic, a 122-page paper solving an open problem sounds great for a Ph.D. thesis. $\endgroup$ 2 days ago
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    $\begingroup$ You might also want to seek advice from someone in an electrical engineering department who is interested in the more theoretical aspects of signal processing. Much of the advice you'll get here on MathOverflow will be from pure mathematicians, and the "culture" of pure mathematics is a little different from the culture of applied mathematics, which in turn is a little different from the culture of electrical engineering. $\endgroup$ 2 days ago
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First of all, I would consider it against the ethics of scientific publishing to accept an offer as a co-author when you were not involved in the research. So I don't think that is viable route.

What you have achieved is quite unusual, you have on your own identified and developed a research direction and produced a set of results that advance the state of the art. Isn't that what a Ph.D. is all about? Rather than seeking a co-author, I would seek a Ph.D. advisor. Contacting an expert in the field, asking for a chance to present your results – with the objective to expand this into a Ph.D. thesis – might very well succeed.

Preparing a seminar in which you present your work would also help you to focus on the essential innovation, which is difficult to extract from the arXiv paper. You might even find that this seminar can be converted into a paper that would be more suitable for publication. In the mean time, by posting your work on arXiv you have established your priority, so a journal publication is not at all urgent.

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  • $\begingroup$ Thank you for your insights Carlo. As per my replies to @TimothyChow, one of the challenges I'd have in going this direction is finding an appropriate expert to be my Ph.D. advisor. Do people sometimes have more than one advisor (I'm thinking a mathematician and electrical engineer in my case)? I'm almost starting to think maybe I could write a research announcement (summary of major results) as suggested by @ SergeiAkbarov to present to experts in the topic of my paper, win them over, and then seek a mathematician for feedback on the technical (mathematical) aspects of the paper. $\endgroup$ 2 days ago
  • $\begingroup$ In essence, my problem is that this paper marries two seemingly disparate fields and I'm not sure how I would proceed. $\endgroup$ 2 days ago
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    $\begingroup$ perhaps my perspective is biased, but I think the merit of your work is the formulation of a useful estimator for conversion gain in photon transfer; the math is secondary, I don't think it is sufficiently interesting for a mathematician; so I would condense the paper so that the body of the text adresses an audience of electrical engineers, and relegate much of chapter 2 to appendices. And again, I think that a helpful way to achieve this condensation is to imagine that you have to give a 30 minute talk on your achievements. $\endgroup$ 2 days ago
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    $\begingroup$ Now, I don't know if anyone works in your precise field in my neighboring department, but a highly mathematical result that solves a problem in engineering would make it a perfect PhD thesis in a department such as MSU-CMSE (The department itself straddles the colleges of natural sciences and engineering.) For students there is it normal to have both mathematicians and engineers on their committees. $\endgroup$ 2 days ago
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    $\begingroup$ The Program in Applied & Computational Mathematics at Princeton University is another Ph.D.-granting program where it is commonplace for a thesis to present a highly mathematical solution to an engineering problem, and thesis committees will have professors from different departments. $\endgroup$ 2 days ago
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You could also just send jour work to a journal, for instance one of those suggested six months ago by your professor. Hopefully, even if they reject it, a referee will tell you what to do in order to improve your work and make it ready for publication, including breaking into more parts. It it true that editing is a very long process. But you would be now 6 months ahead in it, had you followed your advisor's advices.

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    $\begingroup$ "a referee will tell you what to do in order to improve your work and make it ready for publication, including breaking into more parts" <---- one would hope. $\endgroup$ 2 days ago
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    $\begingroup$ one thing to keep in mind is this: you can only publish once; so if you think that your work will benefit by a more compact, focused presentation (which I think is the case), you don't want to first publish a less optimised paper. $\endgroup$ 2 days ago
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    $\begingroup$ @CarloBeenakker actually there is a possibility to publish a research announcement mathoverflow.net/questions/210867/… (and after that the original paper with the complete proof). $\endgroup$ 2 days ago
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    $\begingroup$ indeed, I was concerned with the inverse order: first publish a lengthy detailed exposition and then follow it up with a compact description; most journals will not allow that. $\endgroup$ 2 days ago
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I usually solve this problem by publishing research announcements. I highlight the main results, write them down (without proofs, but with the reference to the original article in arxiv.org) in some small article and send it to a journal where they publish this (there are some Russian and French journals where mathematicians can publish texts like this).

But this is fatally important only if your employer requires you to publish regularly. If this is not necessary, you can simply send your big article to a journal which publish big texts, and if all is well, they will publish it (there are several journals for this, in particular, Journal of Mathematical Sciences). Of course, you will have to wait, most likely several years, but in any case, long articles are published with long delays. And of cource, no one bothers you to simultaneously publish a research announcement, as I wrote above.

Offering co-authorship to an outsider is an absurd idea.

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  • $\begingroup$ Thank you for your answer Sergei. Are you aware of researchers submitting research announcements to arXiv? After some digging on this site, I'm getting the impression that announcements are beginning to fall out of style (due in part to arXiv). Is this perception of mine accurate? Maybe submitting a very condensed summary of results to arXiv would help the relevant researchers see what I've done without having to search through the larger (original) paper? $\endgroup$ 2 days ago
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    $\begingroup$ @Aaron I actually did not hear about announcements right in arxiv. Formally I think this is possible, but I don't see the necessity: in arxiv you can publish the full text with the complete proofs without problems, so why publish an announcement? For me announcements in journals are important for reports for my university of what I was doing in this or that period, that is I why I publish them sometimes (my university does not count publications in arxiv, only in journals). Also they can be useful as a justification of the value of your work. $\endgroup$ 2 days ago
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    $\begingroup$ What you say about condenced summary is reasonable (and important), but I usually write this summary right in the introduction of the paper. $\endgroup$ 2 days ago

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